Equivariant Singular Homology and Cohomology

Let G be a topological group. By a G-space X we mean a topological space X together with a left action of G on X. Under the assumption that G is a compact Lie group, a discrete group or an abelian locally compact group, we constructed in [5] (see also [6]) an equivariant homology and cohomology theory, defined on the category of all G-pairs and G-maps, which both satisfy all seven equivariant Eilenberg-Steenrod axioms and which have the given covariant coefficient system k and contravariant coefficient system m, respectively, as coefficients. We shall here describe a simpler construction of such equivariant homology and cohomology theories which also is valid for an arbitrary topological group G. This simpler construction is described in [6] although main use is still made of the original more complicated construction. We shall also discuss some further properties of these equivariant homology and cohomology theories. For actions by discrete groups equivariant cohomology and homology theories of this type exist before (see G. Bredon [1], [2] and Th. Bröcker [3]). I have been informed by A. Wasserman that S. Willson has also observed that my original construction can be modified to give the simpler one described below.